Under \(H_0\), the distribution of interest and its asymptotic behavior has a very different pattern. It has again a support in \([0,1]\)  a symmetry around \(0.5\) , but it seems to converge to a U-shape distribution with peaks closer to the extremes as \(t \rightarrow \infty \)  (see Figure REF ). Asymptotically, the TS allocation probabilities do not concentrate, and in the specific case of step \(t=1\) , it has a uniform distribution in \([0,1]\)  [1]}.
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